Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Abstract

Let m, n ≥ 2, m ≤ n. It is well-known that the number of (twodimensional) threshold functions on an m × n rectangular grid is t(m, n) = 6 π 2 (mn) 2 + O(m 2 n log n) + O(mn 2 log log n) = 6 π 2 (mn) 2 + O(mn 2 log m).We improve the error term by showing that t(m, n) = 6 π 2 (mn) 2 + O(mn 2 ).

Asymptotics of the number of 2-threshold functions

“…The known asymptotic formulas for the number of threshold functions ( [18,1,12]) are based on their relation to (oriented) prime segments.…”

“…Configurations of planar lattice points inside a circle are also considered in [15,16] and configurations of lattice points inside a sphere are considered in [24]. Configurations produced by straight lines are also intensively studied [5,18,12,2,13,20]. This is partially because of their applications in digital geometry and computer graphics.…”

A characterization of 2-threshold functions via pairs of prime segments

“…A useful characterization of two-dimensional threshold functions via oriented prime segments (i.e., ordered pairs of adjacent integer points) was provided in [25]. In that and the subsequent works [2,19] the relation between threshold functions and prime segments was utilized to estimate the number of threshold functions asymptotically. It is important to note that the endpoints of the segment defining a threshold function f , are essential for f , and hence, f can be defined by an ordered pair of adjacent essential points.…”

A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry

“…We notice that the Farey diagram of order (n, n) has the same degree as the resonance diagram of order n. The asymptotic behaviour of the two different structures only differs by a factor. There are some similarities between (m, n)-cubes, that we redefine below, and threshold functions on a two-dimensional rectangular grid, for which an asymptotic value for the cardinality of these functions has been derived in [11]. And Farey diagrams are used, since long time in computer science: for example, they are also used when we study the preimage of a discrete piece of plane in discrete mathematics, and the Farey diagram for discrete segments were studied by McIlroy in [19].…”